How to add and play video in the 3d viewport?
Internal

Add the object.
Add a material.
Set the material to shadeless
Go to the texture tab and add a texture.
Load the movie as your texture.
Upon loading it enable Auto Refresh and press Match Movie Length once. The length should be updated.
In the 3D viewport, switch to textured or material shading mode.
In the shading options (Properties Panel N), select GLSL.



In the file you linked, the movie is playing, but you have set mapping to UV without specifying a UV Map (the object is not unwrapped). This part doesn't have anything to do with playing a movie, but with UV mapping.
Set the mapping to another type or create a UV map.